You can indeed, that’s how a sports injury attorney makes a living, and you can check this site for more details https://www.aswllp.com/sports-recreation-injuries/. To summarize what that page says, compensation is possible, but it will depend on the details of what happened and how.
For example, it is common for official competitions to require that players sign waivers so they won’t be liable for any injuries suffered during a match. And while those wavers free organizers from a lot of responsibility, you can still seek compensation for some types of injury even if you have signed a waiver. It depends on the details of the injury and how it happened, as well as whether or not it was caused as a natural consequence of the sport in question or it was due to negligence on the part of the organizers.
One good example is a case where another player accidentally kicked the soccer ball into your face. That can cause serious damage, but it is a common risk associated with the sport. If, however, you slipped and hurt your knee because the grass wasn’t properly kept, that may be considered a failure on the part of the organizers. Depending on the details of the case.